So, with the Hall of Heroes, everyone is complaining that Basil is pretty much useless. Why not give him something that'll make people WANT to go into the Clockworks, as well as being a crownsink?

Accessory tickets and an accessory alchemy machine.

[WARNING: WALL OF TEXT AHEAD. READER DISCRETION (those who have attention spans above 2 minutes) ADVISED]

Here's how it'll work:
Basil will sell "Accessory Tickets" based on whatever tier you're in. Like the higher-star recipes, he's only got one in stock of each, and sells 4 different ones determined randomly.
Tier 1: Basic Ticket - 4,000 Crowns
Tier 2: Style Ticket (Dusky Ticket, Heavy Ticket, etc.) and Attachment Ticket (Helmet Top Ticket, Armor Back Ticket, etc.) - 8,000 Crowns
Tier 3: Rare Style Ticket (Divine Ticket, Volcanic Ticket, etc.) and Accessory Ticket (Side Blade Ticket, Maid Headband Ticket, etc.) - 24,000 Crowns. Aura Tickets are considered Accessory Tickets.

The Accessory Alchemy Machines will be in Clockwork Terminals, too. They work like this:
You need 2 tickets: either 2 basic tickets, or a (Rare) Style Ticket and an Accessory or Attachment Ticket. Depending on what Tickets you put in, the material and crown requirements change. The energy cost is always 25. They are never able to craft Crests or the promo items like Dragon or Valkyrie Wings.

2 Basic Style Tickets will need shards, some 1 star and 2 star materials, all chosen by the user in an interface similar to the Trading interface. The crafting cost is 5,000 crowns, and both the accessory and style is determined completely randomly. It has no chance at all of becoming Divine, Volcanic, or Prismatic style nor Halos, Glow-Eyes, or Auras.

A Style and Basic Ticket or Attachment and Basic Ticket will need up to 3 star materials, determined by the Style or Attachment ticket used, as in Toasty Tickets would need Torchstones, while Armor Back Tickets would need Fine Fabric or Drake Scales. Attachment Tickets won't ever craft Halos or Glow-Eyes. Crafting cost will be 7,500 crowns. Military and Toasty Tickets can't craft Parrying Blades, but other than that, Style/Basic will craft a random accessory of that style and Basic/Attachment will craft a random accessory of that attachment and a random style.

Style and Attachment Tickets will need up to 3 star materials, based on what tickets are given. Crafting cost is 10,000 crowns. It crafts an accessory of the Tickets' style and attachment.

A Rare Style and Basic Ticket or Accessory and Basic Ticket will need up to 4 star materials, yet again based on the tickets used. You can't use this ticket combination to craft Halos or Glow Eyes. An Aura and Basic Ticket is the only way to craft an Aura. Costs 15,000 crowns. Do I really have to explain how the crafted accessory is determined?

A Rare Style and Attachment Ticket or Style and Accessory Ticket will need up to 4 star materials, based on (you guessed it) what tickets are used. You can't use this combination to craft Halos, Glow Eyes, or Auras. Costs 17,500 crowns. You should be seeing a pattern on how accessories are determined, but anyway Rare Style/Attachment will craft a random accessory of that style and attachment. See where I'm going with this?

A Rare Style and Accessory Ticket will need up to 5 star materials, based on what you put in etc. etc. etc. You can only craft Halos and Glow Eyes this way. Costs 20,000 crowns. You know the drill for how the accessory is determined.

Now, this may seem like an absurd amount of crowns to pay for a cosmetic item, but think of how much it costs to get a UV. Throw in the crowns you might want to spend re-rolling it to an ASI max and this won't seem as much in comparison.

TOTAL COSTS (for what I would think are common combinations used)
Completely random accessory: up to 2 star materials; 13,000 crowns; 25 energy (Plus expeditioning costs)
Accessory of a chosen style and attachment: up to 3 star materials; 26,000 crowns, 25 energy.
Specific accessory of a chosen style: up to 4 star materials; 49,500 crowns, 25 energy.
Halos and other specific divine, prismatic, or volcanic accessories: up to 5 star materials; 68,000 crowns, 25 energy.

Now compare those to re-rolling 2 UVs. Not as much, right?

TL;DR: Go read the dang thing, it's less time than rereading any research paper you write for school.